232 REV. T. P. KIRKMAN ON THE PARTITIONS 
Proceeding exactly as in the preceding article with 
k&-+1 instead of & (except in the symbols I?(r,4+1) and 
R*(r,4+1)) we arrive at the result, that when / is even, 
P(r, &+1) 
2] Ae ire i eesti 
7 —2k r i: 
era a 
hb 
where <0, v<tr—2k-6, xf$r-k-6; and, as before, 
irreducible fractions are to be reckoned zero, i.e. 7, # and 
& are all even. 
15. The enumeration of the classes R?(r,4+1), R(r,4+1) 
and I?(r,4+1) being effected in terms of 7 and f, we readily 
obtain that of the asymmetrical and most numerous class 
I(r,k+1) of r-gons having two marginal faces and £+1 
diagonals, by the formula of Art. 5, which gives us 
r—a1—k—4 | —1 _ Q\r—2#-3 | -1 
Ate k+1) =1{3,0 . a  Qrk—r+w+4 
— (R°+2R+422)(7,8+1)}, 
where #<0, v<tr—2h-4, xpr—-k-4. 
This is the number of asymmetrically partitioned 7-gons 
having 4+1 diagonals and two marginal faces. The 
numbers denoted by R*, R and IT are given by the 
formule following, which it may be useful to collect into 
one view. 
R*(r, &+1) 
onli ay ae 
iat, et AN 4 (ee . r —2ky* 
aoe oa gp paisa 
= See oe IE eee pe Sites gets So (Art. 6). 
1 2 | 1 2 [2 pl 
This is the number of r-gons partitioned by +1 dia- 
gonals, so as to have two marginal faces and two axes of 
